I think that the title is self-explanatory but I'm thinking about mathematical subjects that have not received a full treatment in book form or if they have, they could benefit from a different approach.
(I do hope this is not inappropriate for MO).

Let me start with some books I would like to read (again with self-explanatory titles)

1) The Weil conjectures for dummies

2) 2-categories for the working mathematician

3) Representations of groups: Linear and permutation representations made side by side

Regarding the Weil conjectures, have you read the appendix to Hartshorne that discusses these? If so, you could also try Nick Katz's exposition on Deligne's work in the Hilbert's Problems book (in the Proceedings of Symposia in Pure Math series) from the 1970s. Also, Deligne's article Weil I is less technical than you might guess, and there is also the textbook by Freitag and Kiehl.
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EmertonJan 24 '11 at 12:44

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Qiaochu: Demazure and Gabriel wrote a book using the functor of points approach over 3 decades ago. Some people love this book, while others...
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Donu ArapuraJan 24 '11 at 17:55

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Maybe there is a place for the dual question: "Books you would like to write (if somebody would just read them)" so people can mention their book ideas and get some feedback.
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Gil KalaiFeb 1 '11 at 15:03

35 Answers
35

I don't know for certain that this doesn't exist, so I'm in a no-lose situation: if this is a rubbish answer then it means that a book that I want to exist does exist. Many mathematicians of a pure bent have taken it upon themselves to get a good understanding of theoretical physics. And many have actually managed this. But it seems to me that they usually go native in the process, with the result that I cease to be able to understand what they are saying. It could be that this is just an irreducibly necessary feature of physics, but I doubt it. Out there in book space I believe there exists a book that explains theoretical physics in a way that physicists would dislike intensely but mathematicians would find much easier to read. It may well be that if you want to do serious work in mathematical physics then you have to understand the subject as physicists do. However, this book would be aimed at pure mathematicians who were not necessarily intending to do serious work in mathematical physics but just wanted to understand what was going on from a distance.

I used to have a similar view about explanations of forcing, but I think Timothy Chow's wonderful Forcing for Dummies has filled that gap now.

Michael Spivak has recently written a book called "Physics for Mathematicians: Mechanics I". I haven't seen it and it's a bit expensive on Amazon, but it might be just what you want (but as far as I can tell it's "only" about classical mechanics...)
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Gonçalo MarquesJan 24 '11 at 15:07

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"Physics for Mathematicians: Mechanics I" is apparently a reworked and expanded version of these notes: math.uga.edu/~shifrin/Spivak_physics.pdf. Now that I know about it, I'm really looking forward to reading it!!! +1
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VectornautJan 24 '11 at 15:36

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+lots. Physics books are usually written in a way that teaches the mathematics through physical intuition... The trouble is that I have no physical intuition. I'd like a book that teaches the physics through mathematical intuition.
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Dylan WilsonJan 24 '11 at 16:32

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Have you read Vladimir Arnold's "Mathematical Methods in Classical Mechanics"? I would say that it fits the bill, but maybe you've read it and it falls short in some way.
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arsmathJan 24 '11 at 16:41

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I really like Folland's book "Quantum field theory, A tourist guide for mathematicians".
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Rob HarronFeb 2 '11 at 2:43

There are many good textbooks in homology and elementary homotopy theory, but the supply of instructive examples they offer is usually appallingly small (spheres and projective spaces are the standard examples, but often there is little beyond). One reason is that to discuss interesting examples, one needs a lot of machinery, whose development consumes time and space.
The books by Hatcher or Bredon offer a lot of examples; and I also like Neil Stricklands bestiary:

I should say that very recently such a book has been written by Audin and Damian - "Théorie de Morse et homologie de Floer", which is a beautiful and comprehensive introduction to the easiest parts of Floer homology. My only complaint with this book is that it doesn't go quite far enough - I guess I'm thinking more of a book the size of McDuff and Salamon's wonderful "J-holomorphic curves and symplectic topology" - but written specifically for Floer theory.
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Will MerryJan 24 '11 at 14:32

Tonny Springer developed a subtle correspondence between Weyl group representations (say over $\mathbb{C}$) and nilpotent orbits of the related semisimple Lie algebra,
showing in particular how to realize the finite group representations in the top cohomology of fibers in his special desingularization of the nilpotent variety.
By now the ideas involved have permeated much of the work in Lie theory due to Lusztig and many other people. But there is no systematic treatise on the subject and its connections with other areas of Lie theory, algebraic geometry, combinatorics. In my 1995 book Conjugacy Classes in Semisimple Algebraic Groups I included toward the end a very short survey of Springer theory, following a treatment of the unipotent and nilpotent varieties. But I realized at the time that I didn't understand the subject deeply enough to write a comprehensive account. (I still don't.)

My first exposure to Springer's ideas unfortunately didn't take hold right away. I recall making a short visit to Utrecht around 1975, where I had lunch with Springer at an Indonesian restaurant and he jotted down the new ideas he was excited about. No napkin or other scrap of paper survives, but anyway I understood only later how amazing his insights were. They deserve a thorough treatment in book form.

Spaces of Diffeomorphisms
For 60+ years this has been a foundation of differential topology, featuring prominently in work of Smale, Cerf, Hatcher, Thurston, and many others; but I don't know any adequate reference. Indeed, it seems only a handful of brilliant people know this stuff, and everyone else uses their work as if it were a collection of black boxes.
My dream book would include, among other things, a modern introduction to Cerf theory from the perspective of Igusa's theory of framed functions, leading up to a readable and self-contained proof of Kirby's Theorem. It would also contain exposition and simplification of theorems of Hatcher, Cerf, Kirby, and Seibenmann.
This is a cheerful prod to a certain prospective author of such a book, that when it is written it will surely become an instant classic; I, for one, will pre-order.

@Maxime. That's a good reference if you want to know about the group theory of Diff, but not if you want to know about its algebraic topology (e.g. what can one say about the homotopy-type of $Diff(S^n)$?). I think Daniel is right that we are missing a book on that topic.
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Tim PerutzJan 31 '11 at 17:03

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@Tim: "I think Daniel is right that we are missing a book on that topic". So do I (I would love this book to explain the links between the cohomology of these groups and foliations, à la Mather-Thurston and the few known results about these cohomology groups).
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Maxime BourriganJan 31 '11 at 17:27

I have in mind the most rigorous modern view, the most intuitive undergraduate calculus view, and the physicist's tensor calculus view. These perspectives can be so different that it's hard to keep in mind that they're all ultimately concerned with the same thing.

Take one concept at a time examine it from a rigorous, intuitive, and computational viewpoint. For example, take a gradient and define it as a differential form, as a vector perpendicular to a surface, and as a tensor. Or here's how a differential geometer, a calculus student, and a physicist all view integrating over a surface. Here's how they each view Stokes' theorem etc.

You should read Spivak's 5 volume "A Comprehensive Introduction to Differential Geometry." In particular, the first 3 volumes. He makes sure to treat almost every single aspect 3 ways: in local coordinates (what you call the physicist's "tensor calculus"), with moving frames (the Cartan/Chern approach), and the modern "invariant" formulation. In my opinion, all differential geometers should be comfortable moving back and forth between all three, because they're all useful in various different situations.
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Spiro KarigiannisJan 24 '11 at 13:14

1

I've read Spivak's 1st volume. I had good intentions of going further but never made it. What I have in mind is a little different from Spivak in that I'd like to see the comparisons from the beginning. Maybe start with geometry from the viewpoint of Schey's book "Div, Grad, Curl and All That" and show how the vast machinery of differential geometry makes these concepts rigorous.
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John D. CookJan 24 '11 at 13:51

I know about Serre's Abelian $\ell$-adic representations and elliptic curves, but I am sure that a more general theory has been established since then. There are a few people who have notes on Galois representations on their web pages, but no book that I know of.

While waiting for Laurent's book (!) , you could try reading Modular Forms and Fermat's Last Theorem (Cornell, Silverman, Stevens eds.), which is a fantastic graduate level introduction to the subject.
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EmertonJan 24 '11 at 12:37

Does such a definition of higher K-groups without topology actually exist? I have never heard about that, so it sounds more like an ambitious research project.
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Johannes EbertJan 25 '11 at 8:54

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Concerning Weyl's The Classical Groups, an argument can be made in favor of either modern text: Goodman & Wallach Symmetry, Representations, and Invariants (2nd ed., Springer GTM 255, 2009) and Procesi Lie Groups (Springer Universitext, 2007). I won't try to make the argument, since what you mean by asking for the same proofs as in Weyl's book might need further discussion.
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Jim HumphreysJan 30 '11 at 18:14

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I don't believe Goodman-Wallach can really supersede Weyl. For example, where are Capelli's identities in Goodman-Wallach? I only see Theorem 5.7.1, which neither gives an explicit form nor applies to the classical case (Goodman-Wallach require $V=S^2(\mathbb C^n)$ or $V=\wedge^2(\mathbb C^n)$, which lead to the Turnbull rsp. Howe-Umeda-Kostant-Sahi identities rather than the actual Capelli ones), let alone an explicit proof "from the definitions". Procesi's text could do the trick indeed.
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darij grinbergJan 30 '11 at 21:25

I know Saunders Mac Lane already wrote a book by that name, but in my opinion his book doesn't live up to its title. His book would perhaps be better named "Category theory for the working algebraist." I'd like to see a book with more examples, especially examples outside of algebra and algebraic topology.

I think Steve Awodey's "Category theory" might be just right for you.
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Gonçalo MarquesJan 24 '11 at 14:42

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Gonçalo: agreed. Awodey is written specifically for computer scientists and other people who don't have much exposure to algebraic topology and the like, so it develops all the necessary examples from scratch (the two most prominent, I think, being posets and monoids).
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Qiaochu YuanJan 24 '11 at 15:55

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Actually, as a non-categorist, I think Mac Lane's title is apt if treated as an introduction to the theory rather than as a handbook for practical reference. But each to their own
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Yemon ChoiJan 24 '11 at 19:32

My answer is quite simple and stupid. I don't know French; so I would like to read EGA, SGA, and BBD in English (or in Russian:)). I also suspect that these books could be updated in the process of translation.:)

I have heard once that Yuri Manin had translated SGA (and EGA?) into Russian. If anyone, he knews if that is correct and if translations of BBD exist too.
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Thomas RiepeJan 26 '11 at 20:34

2

... and, of course, French is a very nice language!
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Thomas RiepeJan 26 '11 at 20:35

5

Mikhail, you will be better off learning the little French required to read EGA, SGA, FGA, BBD, DPP, GAGA, SAGA, etc., than waiting for English or Russian translations to appear.
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Chandan Singh DalawatJan 27 '11 at 5:03

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The problem is that these books would be complicated reading for me even in English.
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Mikhail BondarkoJan 27 '11 at 10:07

The algebraic and differential geometry and Hodge theory side of complex geometry is well established in many books, but I've had real trouble finding examples that are worked out in detail (which would be perfect as exercises, perhaps if given with hints) that show how the theory works in practise and provide counterexamples to some implications. For example, an ample line bundle does not have to admit any global sections, but I've never seen an example of such a bundle given in a textbook.

"Faltings explained" : Several of his articles are very hard to read and existing surveys on his concepts don't really fill the gap. I would like to read a book about his work, his themes, background ideas and techniques which is a readable walk through all that, something like Connes' "NCG"-book + Connes/Marcolli's "noncommutative garden".

"Morava explained" : The same as above on Morava's work, containing a (for the arithmetic geometry inclined reader) readable description of the homotopy theory background. With comments from Manin, Kontsevich and Connes, and a (sci-fi ?) chapter on how homotopy theory and number theory may mutually interfuse (e.g. through "brave new rings").

Mumford suggested in a letter to Grothendieck to publish a suitable edited selection of letters by Grothendieck to his friends, because the letters he received from him were "by far the most important things which explained your ideas and insights ... vivid and unencumbered by the customary style of formal french publications ... express(ing) succintly the essential ideas and motivations and often giv(ing) quite complete ideas about how to overcome the main technical problems ... a clear alternative (to the existing texts) for students who wish to gain access rapidly to the core of your ideas". (Found in the very beautifull 2nd collection)

There are precisely two books on Arakelov geometry. One by Lang and one by Soule. I would love to see a book written on the subject which focuses mainly on the two dimensional (and one-dimensional) case. Sections 8.3 and 9.1 of Liu's book do this greatly for example (but considers only intersection multiplicities at the finite points). It should include all the theorems done so far. Something like

Chapter 0. Prerequisites

Chapter 1.
Arithmetic curves (Riemann-Roch, slopes method, etc. One should include a paragraph or appendix on algebraic curves stating all the theorems that can and have been generalized.)

(N.B. An arithmetic curve is the spec of a ring of integers.)

Chapter 2.
Arithmetic surfaces (This would contain all the "arithmetic" analogues of the theorems mentioned in the Appendix. For example, there has been a lot of work on Riemann-Roch theorems, trace formulas, Dirichlet's higher-dimensional unit theorem, Bogomolov inequalities, etc. Also, there are four intersection theories (which are compatible) I know of at the moment. The one developed by Arakelov-Faltings, then Gillet-Soule, then Bost and then Kuhn. The book should include a detailed description of them.

Appendix A.
Algebraic surfaces. (A survey of all the classical theorems for algebraic surfaces that have an analogue in Arakelov geometry. This includes Faltings' generalizations of the Riemann-Roch theorem, Noether theorem, etc. but also the theorems generalized to Arakelov theory by Gasbarri, Tang, Rossler, Kuhn, Moriwaki, Bost, etc.)

The title is supposed to suggest that this would be about A^1 homotopy theory... starting from scratch and assuming roughly zero exposure to algebraic geometry. Just an answer to the question, "How can we think about algebraic geometry from the perspective of homotopy theory?"
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Dylan WilsonJan 24 '11 at 10:34

One of the great things about this question is that it secretly allows us to ask/answer a bunch of questions of the form "Is there a book like blah about bleh?" Thanks Sean, that book looks great!!
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Dylan WilsonJan 25 '11 at 4:07

Remark: Several items below refer to the formalism of locales. Although consistent usage of the language of locales allows one to get rid of the axiom of choice in almost all cases, my main reasons for it are purely pragmatic: The formalism of locales allows one to obtain equivariant and family versions of many theorems without any additional effort, as opposed to the formalism of topological spaces (think of Hahn-Banach theorem, for example).

A general topology textbook written in the language of locales, with no mention of topological spaces.

Textbooks on commutative algebra and algebraic topology written in the language of locales.
In particular, such textbooks can usually avoid mentioning maximal ideals, the axiom of choice, or Zorn's lemma.

Textbooks on algebraic topology and homological algebra written in the language of (∞,1)-categories.

Higher categories for the working mathematician. This book should contain a lot of examples
of higher categories that are actually used in mathematics outside of category theory.
(For example, the bicategory of algebras, bimodules, and intertwiners,
the tricategory of conformal nets, defects, sectors, and morphisms of sectors etc.)

A textbook on topological vector spaces (in particular, on locally convex, Banach, and nuclear spaces)
written from the categorical viewpoint.
For example, such a textbook would define a nuclear morphism as a morphism
that can be factorized in a certain way (see a recent paper by Stephan Stolz and Peter Teichner).
The textbook should consistently use the language of locales. For example, this allows one to prove Hahn-Banach, Gelfand-Neumark, or Banach-Alaoglu theorems without using the axiom of choice.

Also: I thought it was acknowledged that while you can (and to some extent, should) set up linear algebra without coordinates and bases and matrices, getting things done in functional analysis rather often needs you to choose bases, etc. (Cf. the difference between categories of Hilbert spaces and categories of RKHS)
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Yemon ChoiJan 24 '11 at 19:23

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@Michael: Could you please be more precise? What kind of theorem or definition do you have in mind?
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Dmitri PavlovJan 26 '11 at 2:42

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@Yemon: One size does not fit all. You and darij seem to subtly imply (or at least this is my feeling when I read your comments) that for any mathematical theory there is the best way to expose it, whereas I am more inclined towards diversity of expositions. Some people (like me) like coordinate-free expositions, while others prefer bases and matrices. There are plenty of linear algebra textbooks written using bases and matrices, but very few or none are written in a coordinate-free way. That's why I included linear algebra in my list.
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Dmitri PavlovJan 27 '11 at 19:05

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@Dmitri: I think I for one have not learned anything "literally in a few minutes". Do you have experimental evidence for this claim? Anyway, I agree with you that most people take a lot longer to learn the abstract approach. But moreover, for many people understanding concrete examples is a necessary route to abstraction. If you're going to teach people about dualizable objects in categories, you can go ahead and teach them about bases and matrices first, I think, without wasting anyone's time.
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Pete L. ClarkJan 31 '11 at 4:24

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Having thought things through a bit more, I wish to affirm the principle that the author of a math book ought not to be required to include any material beyond that which is of firm personal interest to herself. (Diligent application of this principle could lead to better books.) So I don't want to discourage anyone from writing this particular take on linear algebra. Rather what I mean to say is that such a book should be used for good rather than ill: raising a generation of mathematicians for whom bases and matrices are no more than an afterthought would be nothing to be proud of.
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Pete L. ClarkJan 31 '11 at 15:46

There are a number of classic books, such as Whittaker and Watson's Modern Analysis, that I'd like to see typeset in TeX and updated slightly. Sometimes notation or terminology have changed and a little footnote would help greatly.

Also by Watson, I'd like to see his 1922 book "A Treatise on the Theory of Bessel Functions" with updated typography and notation. A scan of the book is available here. Apparently the book has entered the public domain and so there would be no legal barrier to producing an updated version.

"We shall now shew..." for instance (W&W, p.13 and many other places.)
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StoppleJan 24 '11 at 20:50

1

Exactly. I had no idea anyone wrote "shew" in the 20th century until I saw that.
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John D. CookJan 24 '11 at 21:44

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But what about Watson's 1944 2nd Edition or the reprinted version: books.google.com/… perhaps these are typographically still similar (or the same) as the 1922 versions, but don't know if the copyright lapse applies anymore?
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SuvritJan 25 '11 at 10:21

Dear Lorenzo, What is your objection to SGA 4.5 (which is my personal favourite of the SGAs)?
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EmertonJan 25 '11 at 3:36

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Dear Emerton, wasn't someone (I think Verdier) originally assigned by Grothendieck the project of replacing the spectral-sequence-laden arguments of SGA4.5 with simpler arguments using derived categories, but it was never finished?
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Harry GindiJan 31 '11 at 15:17

There are currently several books on Lie theory which take a very concrete approach, containing many examples (e.g. Rossmann, Hall, Stillwell). Basically they can be read by a student with some knowledge in calculus, linear algebra and perhaps some mathematical maturity. However, I have yet to find a book on the theory of (linear) algebraic groups which doesn't delve into topics from commutative algebra and algebraic geometry before even defining what an algebraic group is, and even then, most texts take a very abstract approach - most proofs seem like general nonsense to me, but maybe that's just because I'm not an algebraist in heart. In any case, I would very like to see a book on the subject which takes a very concrete approach through examples and constructive proofs.

Some might be surprised that anyone has the desire to read Bourbaki at all ;-)
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Johannes HahnJan 27 '11 at 14:42

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@Chandan: I honestly think that my current musings about "abstract algebraic number theory" are highly in the spirit of the text you name above. For instance, one of the points is the generalization of the Dirichlet Unit Theorem to a wider class of rings, and in this regard there is indeed a Samuel Unit Theorem. In general, Samuel's little book on the algebraic theory of numbers often feels like a little coda to Bourbaki. There are exceptions: for some reason I feel confident that Nicolas would die before mentioning the Minkowski Convex Body Theorem. (Perhaps he has.)
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Pete L. ClarkJan 30 '11 at 21:22

I think Hilbert's 23 problems form an organizatory framework for mathematics, that is much more organic than say the AMS classification. I believe that a book that traces the mathematics that grew from these problems can help to organize the burgeoning state mathematics is currently in.

I'm aware there is a book called "The Honours Class" that gives a history of Hilbert's problems up to their solution. However, this book is more biography than mathematics. Also, I'm interested in what happens after the problem is solved. A case study is the 17th problem, which lead to much of real algebra and real algebraic geometry today.

Dear David, This exists in textbook form: as I noted in another comment, there is the book Modular forms and Fermat's Last Theorem (Cornell, Silverman, Stevens eds.).
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EmertonJan 24 '11 at 12:46

1

There are also the DDT notes, now available on Darmon's website, along with other related material.
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Chandan Singh DalawatJan 24 '11 at 14:16

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Emerton, I have this book but unfortunately haven't had the time to dive into it yet. My impression (horribly mistaken?) was that the last fifteen years have seen some simplifications and improvements to the proof - e.g. appeal to base change to avoid level lowering, appeal to Jacquet-Langlands to study the Hecke algebras in a more hands-on way, the Diamond-Fujiwara version of patching and concomitant avoidance of appeal to multiplicity one, etc.
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David HansenJan 25 '11 at 16:53

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Dear David, Yes, but these improvements are amply documented in the research literature; I don't see the need for another text at the moment, given the existence of Cornell, Silverman, and Stevens. After all, the paper of Diamond in Inventiones is well-written, so if one understands everything in Cornell, Silverman, and Stevens except the mult. one statements, it is no trouble to modify things so as to incorporate the results of Diamond's article. As for replacing the geometric arguments for level lowering by base change, this is very powerful in those contexts where one doesn't have ...
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EmertonJan 27 '11 at 4:33

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... the same tight control of the geometry as one has in the context of modular curves, but it's a matter of one's predilections as to whether it counts as a simplification. (This comment just reflects my own training, which finds Ribet's arithmetic geometry arguments quite a bit easier to follow than the proof of base change.) I think that, with the sole exception of Diamond's paper, which really does count as an unambiguous simplification, these other approaches to the argument just reflect modifications of technique in order to ...
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EmertonJan 27 '11 at 4:39